/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 48 Find the zeros of the function a... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 3: Problem 48

Find the zeros of the function and state the multiplicities. \(d(x)=[x-(2-\sqrt{11})][x-(2+\sqrt{11})]\)

Short Answer

Expert verified
The zeros are \(2 - \sqrt{11}\) and \(2 + \sqrt{11}\), both with multiplicity 1.

Step by step solution

01

Understand the problem

The task is to find the zeros of the function given by the expression and also state their multiplicities. The function provided is in its factored form.
02

Set the function to zero

To find the zeros of the function, set the expression \(d(x)=[x-(2-\sqrt{11})][x-(2+\sqrt{11})]\) equal to zero: \[ d(x) = [x-(2-\sqrt{11})][x-(2+\sqrt{11})] = 0 \]
03

Solve each factor for zero

Set each factor of the function to zero and solve for \(x\).\( x - (2 - \sqrt{11}) = 0 \) adds: \[ x = 2 - \sqrt{11} \]\( x - (2 + \sqrt{11}) = 0 \) adds:\[ x = 2 + \sqrt{11} \]
04

Determine multiplicities

Each zero appears only once in the factored form of the function, which implies that both zeros have a multiplicity of 1.

Unlock Step-by-Step Solutions & Ace Your Exams!

  • Full Textbook Solutions

    Get detailed explanations and key concepts

  • Unlimited Al creation

    Al flashcards, explanations, exams and more...

  • Ads-free access

    To over 500 millions flashcards

  • Money-back guarantee

    We refund you if you fail your exam.

Over 30 million students worldwide already upgrade their learning with 91影视!

Key Concepts

These are the key concepts you need to understand to accurately answer the question.

factored form
In mathematics, the factored form of a polynomial is an expression where the polynomial is written as a product of its factors. This form makes it easy to identify the zeros, or roots, of the polynomial. For example, if we have a function represented in its factored form as \(d(x)=[x-(2-\sqrt{11})][x-(2+\sqrt{11})]\), we immediately see the factors of the function. Factored forms are especially convenient for solving equations and understanding the algebraic structure of functions. By setting each factor to zero, you can find the zeros of the function quickly.
zeros of a function
The zeros of a function are the values of \(x\) that make the function equal to zero. In other words, they are the solutions to the equation where the polynomial equals zero. To find the zeros, we set each factor of the polynomial equal to zero and solve for \(x\). For example, given the function \(d(x)=[x-(2-\sqrt{11})][x-(2+\sqrt{11})]\), we set each factor equal to zero:
\[ x - (2 - \sqrt{11}) = 0 \] \[ x - (2 + \sqrt{11}) = 0 \]
Solving these equations, we get the zeros: \[ x = 2 - \sqrt{11}, \] \[ x = 2 + \sqrt{11}. \]
These are the values where the function touches or crosses the x-axis.
multiplicity
Multiplicity refers to the number of times a particular zero appears in the factored form of a polynomial. If a zero appears more than once, it has a higher multiplicity. In our example, the zeros of the function \(d(x)=[x-(2-\sqrt{11})][x-(2+\sqrt{11})]\) are \(2 - \sqrt{11}\) and \(2 + \sqrt{11}\). Each of these zeros appears exactly once, which means they each have a multiplicity of 1.
Understanding multiplicity is crucial:
  • A zero with a multiplicity of 1 will cross the x-axis at that point.
  • A zero with a multiplicity greater than 1 will touch but not cross the x-axis, creating a 'bounce.'
Recognizing multiplicity helps in graphing the polynomial and predicting its behavior around the zeros.

One App. One Place for Learning.

All the tools & learning materials you need for study success - in one app.

Get started for free

Most popular questions from this chapter

The average round trip speed \(S\) (in mph) of a vehicle traveling a distance of \(d\) miles in each direction is given by $$S=\frac{2 d}{\frac{d}{r_{1}}+\frac{d}{r_{2}}}$$ where \(r_{1}\) and \(r_{2}\) are the rates of speed for the initial trip and the return trip, respectively. a. Suppose that a motorist travels \(200 \mathrm{mi}\) from her home to an athletic event and averages \(50 \mathrm{mph}\) for the trip to the event. Determine the speeds necessary if the motorist wants the average speed for the round trip to be at least \(60 \mathrm{mph}\) ? b. Would the motorist be traveling within the speed limit of \(70 \mathrm{mph} ?\)

A monthly phone plan costs \(\$ 59.95\) for unlimited texts and 400 min. If more than 400 min are used the plan charges an addition \(\$ 0.45\) per minute. For \(x\) minutes used, the average cost per minute \(\overline{C_{1}}(x)\) (in \$) is given by $$\overline{C_{1}}(x)=\left\\{\begin{array}{ll} \frac{59.95}{x} & \text { for } 0 \leq x \leq 400 \\ \frac{59.95+0.45(x-400)}{x} & \text { for } x>400 \end{array}\right.$$ a. Find the average cost if a customer talks for \(252 \mathrm{~min}\) \(366 \mathrm{~min},\) and \(400 \mathrm{~min} .\) Round to 2 decimal places. b. Find the average cost if a customer talks for \(436 \mathrm{~min}\) \(582 \mathrm{~min},\) and \(700 \mathrm{~min}\). Round to 2 decimal places. c. Suppose a second phone plan costs \(\$ 79.95\) per month for unlimited minutes and unlimited texts. Write an average cost function to represent the average cost \(\overline{C_{2}}(x)\) (in \$) for \(x\) minutes used. d. Find the average cost \(\overline{C_{2}}(x)\) for the second plan for \(252 \mathrm{~min}, 400 \mathrm{~min},\) and \(700 \mathrm{~min}\)

Explain why \(f(x)=5 x^{6}+7 x^{4}+x^{2}+9\) has no real zeros.

Use a variation model to solve for the unknown value. The cost to carpet a rectangular room varies jointly as the length of the room and the width of the room. A 10-yd by 15 -yd room costs \(\$ 3870\) to carpet. What is the cost to carpet a room that is 18 yd by 24 yd?

The intensity of radiation varies inversely as the square of the distance from the source to the receiver. If the distance is increased to 10 times its original value, what is the effect on the intensity to the receiver?
See all solutions

Recommended explanations on Math Textbooks

Calculus

Read Explanation

Discrete Mathematics

Read Explanation

Geometry

Read Explanation

Logic and Functions

Read Explanation

Decision Maths

Read Explanation

Applied Mathematics

Read Explanation
View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

Study anywhere. Anytime. Across all devices.